Find the exact value of the given expression. cos(2tan^(-1)((12)/(5)))

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Find the exact value of the given expression.   cos(2tan^(-1)((12)/(5)))

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Find θ: We need to find the exact value of the expression cos(2tan^(-1)(12/5)). To do this, we will use the double angle formula for cosine, which is cos(2θ) = cos^2(θ) - sin^2(θ). However, we first need to find cos(θ) and sin(θ) where θ = tan^(-1)(12/5).Calculate Hypotenuse: Let's denote θ = tan^(-1)(12/5). Since tan(θ) = opposite/adjacent, we can consider a right triangle where the opposite side is 12 and the adjacent side is 5. To find the hypotenuse, we use the Pythagorean theorem: hypotenuse^2 = opposite^2 + adjacent^2.Find cos(θ) and sin(θ): Calculate the hypotenuse using the Pythagorean theorem: hypotenuse^2 = 12^2 + 5^2 = 144 + 25 = 169. Therefore, the hypotenuse is √169 = 13.Apply Double Angle Formula: Now we can find cos(θ) and sin(θ). Since cos(θ) = adjacent/hypotenuse and sin(θ) = opposite/hypotenuse, we have cos(θ) = 5/13 and sin(θ) = 12/13.Calculate cos(2θ): Using the double angle formula for cosine, cos(2θ) = cos^2(θ) - sin^2(θ), we substitute cos(θ) and sin(θ) into the formula: cos(2θ) = (5/13)^2 - (12/13)^2.Combine Fractions: Calculate cos(2θ) by squaring the values: cos(2θ) = (25/169) - (144/169).Final Result: Combine the fractions to find cos(2θ): cos(2θ) = (25 - 144)/169 = -119/169.The exact value of cos(2tan^(-1)(12/5)) is therefore -119/169.

$48$ . Find the exact value of the given expression. $\newline$ $\cos \left(2 \tan ^{-1} \frac{12}{5}\right)$

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$48$ . Find the exact value of the given expression. $\newline$ $\cos \left(2 \tan ^{-1} \frac{12}{5}\right)$